Metaheuristic shuffle

Question

I am currently working on a NP-complete problem, and have implemented a personal genetic algorithm for this purpose. The results are more than I could have expected. With a well-designed fitness function and a couple population/mutation carefully tuned, I guess GA can be an excellent tool in certain cases.

Anyway, I am now looking for a metaheuristic (GA, simulated annealing...) capable of producing an optimal shuffling output.

By shuffle, I mean, in this context, an unbiased (à la Fisher-Yates) random permutation of a finite set. Like a card deck. A huge one (~ 500! permutations).

The values of this set are all different. No collisions are to be expected.

Because of this contraint, I have some difficulties to implement a GA solution. Indeed, the shuffled values cannot be used as genes. It is easy to see why:

#include <iostream>
#include <vector>
#define SPLICING 50 // 50|50 one-point crossover
int crossover(int gene, int DNA_length, int A, int B)
{if (gene < (SPLICING*DNA_length)/100) return A; else return B;}

int main() {
    std::vector<int> A, B, C;
    A = { 3, 4, 8, 12, 2, 0, 9, 7, 10, 20 };
    B = { 8, 10, 3, 4, 20, 0, 7, 9, 2, 12 };
    int DNA_length = int(A.size());
    for (int i=0; i<DNA_length; i++) {
        C.push_back(crossover(i, DNA_length, A[i], B[i]));
                    if (i == DNA_length/2) std::cout << "| ";
                    std::cout << C[i] << " ";}
            }

Output: 3 4 8 12 2 | 0 7 9 2 12

There are two collisions (2, 12).

My expected output is something like this: 3 4 8 12 2 | 0 7 9 10 20 (no collision, perfect shuffle of the original set).

Then, I need to encode the order of these values in order to avoid this kind of difficulties.

A naive way is to identify each value with a unique key. But the set then created is an ordinal one because it refers to the sequencing of the values.

I appears that the crossover function has to deal with the ordinality of the DNAs of the parents. But I can not wrap my head around the issue of mixing two nonlinearly ordered ordinal subsets (parents' DNA slices) of an ordinal set (whole DNA) without collision!

Maybe I can rely only on mutation for convergence. No selection, no parents/children, and only a swap function in the same set (individual's DNA). In short: not very convincing.

It is indeed easy to permute ordinal numbers in a unique finite set (e.g., trivially: the first becomes the seventh; the second, the tenth, etc.). But I am not sure if it makes sense to says that the first of set A becomes the seventh when the second of set B becomes the tenth of the new set.

Then, my question is:

In your opinion, can the ordinality of a set be shuffled using a crossover function in the context of a genetic algorithm? If no, can you suggest a metaheuristic approach more efficient for this purpose than a brute-force, hill climbing technique or genetic algorithm?

Thank you.

Answer 1

What you are looking for is called order based genetic algorithms. You have many order based crossover and mutation operators that are meant to work with this kind of problem. The simplest crossover operator works as follows:

1. Select to crossover points
2. Copy the part from parent1 within the crossover points to the first son.
3. Make a list of the elements of parent1 that are outside the crossover points
4. Put the elements of the unused list in the same order of that used in parent2
5. Copy those elements to the first son in the order established in step 4.

You can see an example from my book in the figure below (sorry, but the descriptions are in portuguese - please correlate to the list above):

You can search the web for order based operators or if you prefer, please check the figures from my book at My Geneetic Algorithm book. The figures that interest you are those from chapter 10 (you can use Google translator to understand the legends).

You do not have to mind that the book uses sequential numbers - if you have no repetition, all the concepts explained are valid to your problem.

I hope it helps.